Planar moments

For simple sulfones, the planar moment of the molecule is determined by the positions of the oxygen atoms, thus the knowledge of the O O distance may considerably aid the assignment of rotational transitions (see more details in Reference 5). 

In mechanics, the inertial properties of a rotating rigid body are fully described by its inertial moment tensor I. We can simplify the subsequent equations if we employ in place of I the closely related planar moment tensor P, apparently first used by Kraitchman [4], At any stage of the calculations, however, an equivalent equation could be given which involves I instead of P. The principal planar moments P (g = x, y, z) are the three eigenvalues of the planar moment tensor P and the principal inertial moments Ig the eigenvalues of the inertial moment tensor I. Pg, Ig, and the rotational constants Bg = f/Ig are equivalent inertial parameters of the problem investigated (/conversion factor). 

The rotational motion of the rigid set of mass points about any axis through its center of mass in the absence of exterior forces is known as the free rotation of the rigid body. The planar moment tensor for this motion, with the position vectors ra referred to an arbitrary basis system, can be compactly written as a dyadic (T denotes transposition) [8,32],... 

The inertial moment tensor I and the planar moment tensor P are related by,... 

The basic equations of the -method will be presented later within the framework of the more general r -fit problem. A rigid mass point model, which is strictly true only for the equilibrium configuration, is assumed. The application of Kraitch-man s equations (see below) to localize an atomic position requires (1) the principal planar moments (or equivalent inertial parameters) of the parent or reference molecule with known total mass, and (2) the principal planar moments of the isotopomer in which this one atom has been isotopically substituted (with known mass difference). The equations give the squared Cartesian coordinates of the substituted atom in the PAS of the parent. After extracting the root, the correct relative sign of a coordinate usually follows from inspection or from other considerations. The number, identity, and positions of nonsubstituted atoms do not enter the problem at all. To determine a complete molecular structure, each (non-equivalent) atomic position must have been substituted separately at least once, the MRR spectra of the respective isotopomers must all have been evaluated, and as many separate applications of Kraitchman s equations must be carried out. 

One of the first computer programs written to obtain the Cartesian atomic coordinates referred to the PAS of the parent by means of a least-squares fit to the inertial or planar moments of a number of isotopomers (also multiply substituted) appears to have been the program STRFIT coded by Schwendeman [6]. It is a versatile r0-type program incorporating many useful features, it is not a rs-fit program in the sense in which this term is used in this paper. 

In a practical case, many of the Ama(s) will vanish (for an atom that has never been substituted, all Ama(s) vanish). The planar moment tensor of the parent, with reference to its own PAS (cf. Eq. 11),... 

Kraitchman s basic idea was to introduce into this first (diagonal) tensor term of Eq. 35 the three experimental principal planar moments of the parent PgWexp(l) as obtained from the MRR spectrum and treat them as independent experimental information. This is the essential distinguishing feature between any retype and any rQ-type method and all differences between the two types of treatment may be traced back to this fact. The first term of Eq. 35 is now written as P p(l). It >s then convenient to replace the notation for the planar tensor P[11(s) (Eq. 35) by I V) to distinguish this new function (Eq. 36) from Eq. 35. Note that Eq. 36, in contrast to Eq. 35, depends explicitly on the positions of only those atoms that have actually been substituted in the isotopomer 5 ... 

Let T(s) be the orthogonal transformation that diagonalizes nI1](s) with eigenvalues II w(s)- Kraitchman equates these eigenvalues to the experimental principal planar moments Pgu 1 exp(s) of the isotopomer s ... 

Suppose that the X-D bond is shorter than the X-H bond by 8r. The uncorrected rs-method would then let the X-H bond appear too short by 28r [57]. After the -coordinates r ] for atom X and preliminary -coordinates for the apparent location r[] of H, have been obtained from the uncorrected planar moments, a hypothetical position H of the hydrogen atom after an intentional corrective X-H bond elongation by 28r (usually chosen near 2 x 0.3 pm) is assumed as,... 

The first and second moment conditions can be very easily introduced into the r5-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete restructure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding r/e-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the r -fit because of substitution on a principal plane or axis, were also omitted from the r/E-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. 

Calculating an effective or r0-structure introduced, for the first time, least-squares fitting into the determination of molecular structure from MRR spectra. Unpublished computer programs for this purpose must have been in use by various groups for a long time. A more widely used program with many options, STRFTQ, has been coded by Schwendeman [6], It uses internal coordinates and fits inertial or planar moments, but also isotopic differences of moments. When the differences Pg exp(.v) - Pg exp( 1) are fitted, the structure is called a pseudo-Kraitchman or p-Kr structure, because these differences are the quantities that also dominate Kraitchman s equations. Therefore, the p-Kr structure should approach the restructure. By the criterion adhered to in the present paper (see preceding section), the p-Kr structure is a r0-type structure. 

In a recent paper [55], the multitude of possible r0-fitting schemes have been ordered under systematic aspects. Any of the three major types of rotational parameters, principal inertial and planar moments, and rotational constants, or isotopic differences of these quantities between differently chosen members of the available substitution set, could be r0-fitted. The basic experimental information evaluated from the MRR-spectrum of any molecular species will usually consist of... 

While the relations between the inertial and planar moments are strictly linear and constant, the relation between the increments of the rotational constants Bg and the moments, say Ig, is a truncated series expression and only approximately linear, Alg = ( f/B2g)ABg. Also, the transformation coefficients f/B2g are not strictly constant (nonrandom), although usually afflicted with only a very small relative error. The approximations are, in general, good enough to satisfy the requirements for Eq. 22, and for the above statement rt = rp = rB to be true for all practical purposes. 

The simple matrix for the linear transformation of, e.g., the planar moments Pg(s) of a SDS into the isotopic differences AsAPg - Pg(s) - / (l) is non-square and hence singular because there are less differences than moments. The transformation of the problem is formally described by Eq. 22, but since the transformation matrix C is now singular, the arguments following Eq. 22 are not applicable. Therefore, symbolically rp, where rAP is the p-Kr structure, which is thereby shown to... 

As discussed in the preceding section, the r5-method (Kraitchman s equations or the rs-fit) is practically insensitive to a (small) change of a planar moment component, as represented by the rovib contributions, if the change is common to both the parent and the isotopomer s, and if the rotation by the diagonalizing matrix T(s) (Eq. 41) is small. Therefore, it usually does not matter whether experimental ground state planar moments F exp(s) or, after the T s have been obtained from a preceding... 

In the following, we present the essential equations of the r/ -fit. They are simply those of the r0-method, with three more variables added. For the sake of a unified development, all relations are expressed in planar moments. The model for the structural part is the rigid set of mass points only when the calculated moments are equated to the experimental ground state moments, are they supplemented by three constant rovib contributions. 

We pause to state that the treatment which led us from Eq. 67 to the matrix X of Eq. 76 is not unique. Instead of Eq. 68, the indirect dependence of the planar moments P m(s) on the internal coordinates 5, could have been expressed by ... 

If the planar moment components, Pg g-x,y,z), and their rovib contributions, T s = f ( x + ev + 8,) - g, are used, the corresponding equations required for the rm-method can be found also for the general polyatomic molecule (asymmetric top). The equations are approximate and hold under the provision that the PAS rotation upon isotopic substitution is small [24],... 

Only in the rare case of very small or highly symmetric molecules will the small number (< 3) of planar moment components P" (1) of the parent alone suffice to compute the molecular restructure. To make available for this purpose the three mass-dependence planar moment components PJ(. ) = 2P (.v) - P (s) of only one additional isotopomer s, all isotopomers which are single-substituted with respect to this isotopomer s must have been measured and evaluated in order to find P is). For all but very small molecules the expense involved is prohibitive. 

The other two principal planar moments and expressions for b and c are given by cyclic permutations of a, b, c. Equations of the form of Eq. (11) are known as the Kraitchman equations. Special cases arise for molecules with various symmetry elements. For substitution of an atom in a linear molecule, the magnitude of the a coordinate of the substituted atom is... 

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